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Properties

Label 7225.2.a.bp
Level $7225$
Weight $2$
Character orbit 7225.a
Self dual yes
Analytic conductor $57.692$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7225,2,Mod(1,7225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7225 = 5^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7225.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(57.6919154604\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 30x^{10} + 343x^{8} - 1860x^{6} + 4823x^{4} - 5230x^{2} + 1681 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{9} q^{3} + (\beta_{6} + 1) q^{4} + ( - \beta_{11} + 2 \beta_{2} - \beta_1) q^{6} - \beta_{5} q^{7} + (\beta_{10} - \beta_{4}) q^{8} + (\beta_{6} + \beta_{3} + 2) q^{9} + ( - \beta_{11} - \beta_{2}) q^{11}+ \cdots + (3 \beta_{11} + 2 \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{4} + 28 q^{9} + 4 q^{16} + 48 q^{19} + 24 q^{21} + 24 q^{26} + 68 q^{36} - 28 q^{49} + 72 q^{59} - 76 q^{64} + 8 q^{66} + 88 q^{69} + 48 q^{76} + 60 q^{81} - 40 q^{84} - 16 q^{86} - 16 q^{89}+ \cdots + 96 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 30x^{10} + 343x^{8} - 1860x^{6} + 4823x^{4} - 5230x^{2} + 1681 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} - 30\nu^{9} + 302\nu^{7} - 1040\nu^{5} - 179\nu^{3} + 4610\nu ) / 820 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{11} + 49\nu^{9} - 86\nu^{7} - 1882\nu^{5} + 8737\nu^{3} - 7311\nu ) / 1640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + 20\nu^{6} - 122\nu^{4} + 220\nu^{2} - 41 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{10} - 25\nu^{8} + 212\nu^{6} - 680\nu^{4} + 571\nu^{2} + 145 ) / 120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{11} - 169\nu^{9} + 1499\nu^{7} - 6173\nu^{5} + 11662\nu^{3} - 3974\nu ) / 2460 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} - 20\nu^{6} + 132\nu^{4} - 320\nu^{2} + 191 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -10\nu^{11} + 259\nu^{9} - 2405\nu^{7} + 9293\nu^{5} - 11125\nu^{3} - 4936\nu ) / 2460 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} + 25\nu^{8} - 227\nu^{6} + 905\nu^{4} - 1456\nu^{2} + 530 ) / 60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\nu^{11} - 349\nu^{9} + 3516\nu^{7} - 16308\nu^{5} + 33343\nu^{3} - 20799\nu ) / 1640 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} - 25\nu^{8} + 222\nu^{6} - 830\nu^{4} + 1201\nu^{2} - 505 ) / 40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -11\nu^{11} + 289\nu^{9} - 2707\nu^{7} + 10743\nu^{5} - 16686\nu^{3} + 7264\nu ) / 820 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} - \beta_{4} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{9} + 9\beta_{7} + 3\beta_{5} - 2\beta_{2} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{10} + 10\beta_{8} + 2\beta_{6} - 10\beta_{4} + 2\beta_{3} + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{11} + 28\beta_{9} + 73\beta_{7} + \beta_{5} - 28\beta_{2} + 61\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 91\beta_{10} + 87\beta_{8} + 30\beta_{6} - 99\beta_{4} + 30\beta_{3} + 275 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 76\beta_{11} + 326\beta_{9} + 605\beta_{7} - 121\beta_{5} - 310\beta_{2} + 551\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 820\beta_{10} + 740\beta_{8} + 356\beta_{6} - 980\beta_{4} + 336\beta_{3} + 2289 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1032\beta_{11} + 3504\beta_{9} + 5153\beta_{7} - 1783\beta_{5} - 3264\beta_{2} + 5017\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7437\beta_{10} + 6285\beta_{8} + 3900\beta_{6} - 9621\beta_{4} + 3400\beta_{3} + 19725 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12168\beta_{11} + 36146\beta_{9} + 44801\beta_{7} - 19981\beta_{5} - 33778\beta_{2} + 45831\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.733961
0.733961
2.40715
−2.40715
1.27039
−1.27039
−2.89588
2.89588
2.05077
−2.05077
3.07592
−3.07592
−2.38621 −3.15462 3.69399 0 7.52757 0.219993 −4.04223 6.95160 0
1.2 −2.38621 3.15462 3.69399 0 −7.52757 −0.219993 −4.04223 6.95160 0
1.3 −1.80333 −0.561698 1.25200 0 1.01293 −3.11199 1.34889 −2.68450 0
1.4 −1.80333 0.561698 1.25200 0 −1.01293 3.11199 1.34889 −2.68450 0
1.5 −0.232389 −2.39435 −1.94600 0 0.556420 −2.06570 0.917007 2.73289 0
1.6 −0.232389 2.39435 −1.94600 0 −0.556420 2.06570 0.917007 2.73289 0
1.7 0.232389 −2.39435 −1.94600 0 −0.556420 −2.06570 −0.917007 2.73289 0
1.8 0.232389 2.39435 −1.94600 0 0.556420 2.06570 −0.917007 2.73289 0
1.9 1.80333 −0.561698 1.25200 0 −1.01293 −3.11199 −1.34889 −2.68450 0
1.10 1.80333 0.561698 1.25200 0 1.01293 3.11199 −1.34889 −2.68450 0
1.11 2.38621 −3.15462 3.69399 0 −7.52757 0.219993 4.04223 6.95160 0
1.12 2.38621 3.15462 3.69399 0 7.52757 −0.219993 4.04223 6.95160 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(17\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.b even 2 1 inner
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7225.2.a.bp 12
5.b even 2 1 inner 7225.2.a.bp 12
5.c odd 4 2 1445.2.b.f 12
17.b even 2 1 inner 7225.2.a.bp 12
17.d even 8 2 425.2.e.d 12
85.c even 2 1 inner 7225.2.a.bp 12
85.g odd 4 2 1445.2.b.f 12
85.k odd 8 2 85.2.j.c ✓ 12
85.m even 8 2 425.2.e.d 12
85.n odd 8 2 85.2.j.c ✓ 12
255.v even 8 2 765.2.t.e 12
255.ba even 8 2 765.2.t.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.j.c ✓ 12 85.k odd 8 2
85.2.j.c ✓ 12 85.n odd 8 2
425.2.e.d 12 17.d even 8 2
425.2.e.d 12 85.m even 8 2
765.2.t.e 12 255.v even 8 2
765.2.t.e 12 255.ba even 8 2
1445.2.b.f 12 5.c odd 4 2
1445.2.b.f 12 85.g odd 4 2
7225.2.a.bp 12 1.a even 1 1 trivial
7225.2.a.bp 12 5.b even 2 1 inner
7225.2.a.bp 12 17.b even 2 1 inner
7225.2.a.bp 12 85.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7225))\):

\( T_{2}^{6} - 9T_{2}^{4} + 19T_{2}^{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{6} - 16T_{3}^{4} + 62T_{3}^{2} - 18 \) Copy content Toggle raw display
\( T_{7}^{6} - 14T_{7}^{4} + 42T_{7}^{2} - 2 \) Copy content Toggle raw display
\( T_{11}^{6} - 32T_{11}^{4} + 16T_{11}^{2} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 9 T^{4} + 19 T^{2} - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - 16 T^{4} + \cdots - 18)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 14 T^{4} + 42 T^{2} - 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 32 T^{4} + 16 T^{2} - 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 22 T^{4} + \cdots - 100)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T - 4)^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} - 72 T^{4} + \cdots - 50)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 94 T^{4} + \cdots - 2312)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 66 T^{4} + \cdots - 6498)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 38 T^{4} + \cdots - 200)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 152 T^{4} + \cdots - 2592)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 66 T^{4} + \cdots - 1444)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 82 T^{4} + \cdots - 13924)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 156 T^{4} + \cdots - 110224)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{12} \) Copy content Toggle raw display
$61$ \( (T^{2} - 32)^{6} \) Copy content Toggle raw display
$67$ \( (T^{6} - 214 T^{4} + \cdots - 93636)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 132 T^{4} + \cdots - 13122)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 152 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 174 T^{4} + \cdots - 114242)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 58 T^{4} + \cdots - 2916)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 4 T^{2} + \cdots - 1590)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} - 518 T^{4} + \cdots - 3645000)^{2} \) Copy content Toggle raw display
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